p-torsion elements in local cohomology modules. II
p-torsion elements in local cohomology modules. II
复制标题
局部上同调模中的 p 扭转元。
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发表时间：
2004
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通讯作者：
Anurag Singh
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作者：
Anurag Singh
Gennady Lyubeznik猜想，如果R是正则环，并且a是R的理想，则局部上同调模H i a（R）只有有限个相关联的素理想，[Ly 1，备注3.7（iii）]。虽然这个猜想在一般性上仍然是开放的，但现在有几个结果：如果正则环R包含一个素特征p > 0的域，Huneke和Sharp在[HS]中证明了H i a（R）的相关素理想的集合是有限的。如果R是一个正则局部环，包含一个特征为零的域，Lyubeznik证明了H i a（R）只有1000个相关联的素理想，参见[Ly 1]和[Ly 2，Ly 3]。最近Lyubeznik也证明了这个结果的非分歧正则局部环的混合特征，[Ly 4]。在[胡]克雷格Huneke首先提出了以下问题：对于诺特环R，一个理想一个R，和一个R生成R-模M，是一些相关的素理想H一个（M）总是有限的？对于一些工作在这个问题上，我们请读者的文件[BL，BRS，他]除了上述提到的。在[Si]中，我们构造了一个超曲面R的例子，其中局部上同调模H3 a（R）对每个素数p都有p-挠元，从而有无穷多个相关联的素理想。由于这是迄今为止唯一已知的无穷多个相关素理想的来源，因此值得研究类似的技术是否可以产生一个正则环R的例子，其中局部上同调模H i a（R）对每个素数p都有p-挠元。这导致了一些非常有趣的问题，我们将在本文中看到。到目前为止，我们的结果支持Lyubeznik的猜想，局部上同调模的所有正规环只有1000多个相关的素理想。设R是整数上的多项式环，Fi，Gi是R的元素，其中
Gennady Lyubeznik conjectured that if R is a regular ring and a is an ideal of R, then the local cohomology modules H i a (R) have only finitely many associated prime ideals, [Ly1, Remark 3.7 (iii)]. While this conjecture remains open in this generality, several results are now available: if the regular ring R contains a field of prime characteristic p > 0, Huneke and Sharp showed in [HS] that the set of associated prime ideals of H i a (R) is finite. If R is a regular local ring containing a field of characteristic zero, Lyubeznik showed that H i a (R) has only finitely many associated prime ideals, see [Ly1] and also [Ly2, Ly3]. Recently Lyubeznik has also proved this result for unramified regular local rings of mixed characteristic, [Ly4]. In [Hu] Craig Huneke first raised the following question: for a Noetherian ring R, an ideal a ⊂ R, and a finitely generated R-module M , is the number of associated primes ideals of H i a (M) always finite? For some of the work on this problem, we refer the reader to the papers [BL, BRS, He] in addition to those mentioned above. In [Si] we constructed an example of a hypersurface R for which a local cohomology module H3 a (R) has p-torsion elements for every prime integer p, and consequently has infinitely many associated prime ideals. Since this is the only known source of infinitely many associated prime ideals so far, it is worthwhile to investigate whether similar techniques may yield an example of a regular ring R for which a local cohomology module H i a (R) has p-torsion elements for every prime integer p. This leads to some very intriguing questions as we shall see in this paper. Our results thus far support Lyubeznik’s conjecture that local cohomology modules of all regular rings have only finitely many associated prime ideals. Let R be a polynomial ring over the integers and Fi, Gi be elements of R for which